1ZHETRD(1) LAPACK routine (version 3.2) ZHETRD(1)
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6 ZHETRD - reduces a complex Hermitian matrix A to real symmetric tridi‐
7 agonal form T by a unitary similarity transformation
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10 SUBROUTINE ZHETRD( UPLO, N, A, LDA, D, E, TAU, WORK, LWORK, INFO )
11 CHARACTER UPLO
13 INTEGER INFO, LDA, LWORK, N
15 DOUBLE PRECISION D( * ), E( * )
17 COMPLEX*16 A( LDA, * ), TAU( * ), WORK( * )
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21 ZHETRD reduces a complex Hermitian matrix A to real symmetric tridiago‐
22 nal form T by a unitary similarity transformation: Q**H * A * Q = T.
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25 UPLO (input) CHARACTER*1
26 = 'U': Upper triangle of A is stored;
27 = 'L': Lower triangle of A is stored.
28 N (input) INTEGER
30 The order of the matrix A. N >= 0.
31 A (input/output) COMPLEX*16 array, dimension (LDA,N)
33 On entry, the Hermitian matrix A. If UPLO = 'U', the leading
34 N-by-N upper triangular part of A contains the upper triangular
35 part of the matrix A, and the strictly lower triangular part of
36 A is not referenced. If UPLO = 'L', the leading N-by-N lower
37 triangular part of A contains the lower triangular part of the
38 matrix A, and the strictly upper triangular part of A is not
39 referenced. On exit, if UPLO = 'U', the diagonal and first
40 superdiagonal of A are overwritten by the corresponding ele‐
41 ments of the tridiagonal matrix T, and the elements above the
42 first superdiagonal, with the array TAU, represent the unitary
43 matrix Q as a product of elementary reflectors; if UPLO = 'L',
44 the diagonal and first subdiagonal of A are over- written by
45 the corresponding elements of the tridiagonal matrix T, and the
46 elements below the first subdiagonal, with the array TAU, rep‐
47 resent the unitary matrix Q as a product of elementary reflec‐
48 tors. See Further Details. LDA (input) INTEGER The leading
49 dimension of the array A. LDA >= max(1,N).
50 D (output) DOUBLE PRECISION array, dimension (N)
52 The diagonal elements of the tridiagonal matrix T: D(i) =
53 A(i,i).
54 E (output) DOUBLE PRECISION array, dimension (N-1)
56 The off-diagonal elements of the tridiagonal matrix T: E(i) =
57 A(i,i+1) if UPLO = 'U', E(i) = A(i+1,i) if UPLO = 'L'.
58 TAU (output) COMPLEX*16 array, dimension (N-1)
60 The scalar factors of the elementary reflectors (see Further
61 Details).
62 WORK (workspace/output) COMPLEX*16 array, dimension (MAX(1,LWORK))
64 On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
65 LWORK (input) INTEGER
67 The dimension of the array WORK. LWORK >= 1. For optimum per‐
68 formance LWORK >= N*NB, where NB is the optimal blocksize. If
69 LWORK = -1, then a workspace query is assumed; the routine only
70 calculates the optimal size of the WORK array, returns this
71 value as the first entry of the WORK array, and no error mes‐
72 sage related to LWORK is issued by XERBLA.
73 INFO (output) INTEGER
75 = 0: successful exit
76 < 0: if INFO = -i, the i-th argument had an illegal value
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79 If UPLO = 'U', the matrix Q is represented as a product of elementary
80 reflectors
81 Q = H(n-1) . . . H(2) H(1).
82 Each H(i) has the form
83 H(i) = I - tau * v * v'
84 where tau is a complex scalar, and v is a complex vector with v(i+1:n)
85 = 0 and v(i) = 1; v(1:i-1) is stored on exit in
86 A(1:i-1,i+1), and tau in TAU(i).
87 If UPLO = 'L', the matrix Q is represented as a product of elementary
88 reflectors
89 Q = H(1) H(2) . . . H(n-1).
90 Each H(i) has the form
91 H(i) = I - tau * v * v'
92 where tau is a complex scalar, and v is a complex vector with v(1:i) =
93 0 and v(i+1) = 1; v(i+2:n) is stored on exit in A(i+2:n,i), and tau in
94 TAU(i).
95 The contents of A on exit are illustrated by the following examples
96 with n = 5:
97 if UPLO = 'U': if UPLO = 'L':
98 ( d e v2 v3 v4 ) ( d )
99 ( d e v3 v4 ) ( e d )
100 ( d e v4 ) ( v1 e d )
101 ( d e ) ( v1 v2 e d )
102 ( d ) ( v1 v2 v3 e d ) where d
103 and e denote diagonal and off-diagonal elements of T, and vi denotes an
104 element of the vector defining H(i).
105 LAPACK routine (version 3.2) November 2008 ZHETRD(1)